Apparatus to reduce pressure and thermal sensitivity of high precision optical displacement sensors

ABSTRACT

Methods, systems and devices for estimating a parameter of interest in a borehole. The method may include generating information from an optical displacement device relating to relative motion between two or more reflective surfaces thereof that is indicative of the parameter of interest; and preventing changes in the information resulting from changes at the optical displacement device in at least one of i) temperature, or ii) pressure, by compensating for the changes. Compensating may include adjusting at least one light source generating an electromagnetic beam at least partly received by the optical displacement device responsive to information relating to a control optical displacement device at the optical displacement device. Compensating may include using an optical displacement device and configuring the optical displacement device such that a difference between a first variable gap and a second variable gap is substantially zero while the apparatus is subject to nominal conditions.

BACKGROUND OF THE DISCLOSURE

1. Field of the Disclosure

In one aspect, this disclosure generally relates methods and apparatusesfor sensing displacement optically.

2. Background of the Art

Displacement sensors, such as microphones and pressure sensors, arewell-known. Many displacement sensors may be based on one or more ofelectrical capacitance, electrical impedance, or magnetic fields. Theseelectrical and magnetic based displacement sensors may be limited due toone or more of: low sensitivity, the need for high-voltage biasing, poorelectrical isolation, environmental factors, and responsenonlinearities. These limitations may require a close coupling betweentransducer design and the sensor mechanical design, which may limitperformance and the operational size of the displacement sensor. Opticaldisplacement sensors, such as displacement sensors using an etalon, maybe electrically and magnetically insensitive, which may mitigatelimitations found in electrical and magnetic based displacement sensors.

SUMMARY OF THE DISCLOSURE

In aspects, the present disclosure is related to an apparatus and methodfor estimating a parameter of interest using values of a beam propertyfrom at least one electromagnetic beam that pass through at least partof an optical displacement device.

One embodiment according to the present disclosure includes an apparatusfor estimating a parameter of interest, comprising: an opticaldisplacement sensor comprising: an optical displacement deviceconfigured to receive at least one electromagnetic beam, the opticaldisplacement device comprising: a first reflective surface; a secondreflective surface; a third reflective surface fixed with respect to thesecond reflective surface; a fourth reflective surface fixed withrespect to the first reflective surface; a first variable gap betweenthe first reflective surface and the second reflective surface; and asecond variable gap between the third reflective surface and the fourthreflective surface; and at least one detector array configured toreceive a part of the at least one electromagnetic beam.

A relative motion between the first reflective surface and the secondreflective surface and a relative motion between the third reflectivesurface and the fourth reflective surface may each be indicative of theparameter of interest. The optical displacement device may comprise amiddle member comprising the second reflective surface and the thirdreflective surface. The middle member may comprise a proof mass. Theproof mass may be opaque. The optical displacement device may comprisean outer member comprising the first reflective surface and the fourthreflective surface. The outer member may comprise a proof mass. Theoptical displacement device may be configured such that the differencebetween the first variable gap and the second variable gap issubstantially zero while the apparatus is subject to nominal conditions.Nominal conditions may include at least one of: i) reference gravity;ii) zero force; iii) zero acceleration; iv) zero pressure. The at leastone electromagnetic beam may comprise a plurality of electromagneticbeams. The apparatus may be configured to direct a first electromagneticbeam towards the first variable gap and a second electromagnetic beamtowards the second variable gap.

Another embodiment according to the present disclosure includes anapparatus for estimating a parameter of interest, comprising: an opticaldisplacement sensor comprising: at least one light source configured togenerate at least one initial electromagnetic beam; an opticaldisplacement device configured to receive at least one electromagneticbeam comprising at least a portion of the initial electromagnetic beam;at least one detector configured to receive a part of the at least oneelectromagnetic beam received by the optical displacement device andprovide information relating to the parameter of interest; and a closedloop light source control comprising an optical component at the opticaldisplacement device, the closed loop light source control configured tocompensate for changes in temperature at the optical displacement deviceby adjusting the at least one light source.

The apparatus may include a housing enclosing the optical displacementdevice and the optical component. The housing may be environmentallysealed. The housing may maintain a vacuum inside the housing. Theoptical component may include a beam splitter configured to divide theat least one initial electromagnetic beam into the at least oneelectromagnetic beam and at least one second electromagnetic beam; and acontrol optical displacement device configured to receive the at leastone second electromagnetic beam. The apparatus may include at least onedetector optically coupled to the optical component and configured toreceive a part of the at least one second electromagnetic beam; andcontrol electronics operatively coupled to the at least one detector andthe at least one light source and configured to provide closed loopcontrol of the at least one light source using information from the atleast one detector.

Another embodiment according to the present disclosure includes a methodfor estimating a parameter of interest in an earth formation intersectedby a borehole. The method may include conveying an optical displacementdevice in the borehole; generating information from the opticaldisplacement device relating to relative motion between two or morereflective surfaces of the optical displacement device that isindicative of the parameter of interest; and preventing changes in theinformation resulting from changes at the optical displacement device inat least one of i) temperature, or ii) pressure, by compensating for thechanges at the optical displacement device. Compensating may includeadjusting at least one light source generating an electromagnetic beamat least partly received by the optical displacement device, theadjusting responsive to information relating to a control opticaldisplacement device at the optical displacement device. Compensating mayinclude using an optical displacement device comprising: a firstreflective surface; a second reflective surface; a third reflectivesurface fixed with respect to the second reflective surface; a fourthreflective surface fixed with respect to the first reflective surface; afirst variable gap between the first reflective surface and the secondreflective surface; and a second variable gap between the thirdreflective surface and the fourth reflective surface; and configuringthe optical displacement device such that the difference between thefirst variable gap and the second variable gap is substantially zerowhile the apparatus is subject to nominal conditions.

Examples of the more important features of the disclosure have beensummarized rather broadly in order that the detailed description thereofthat follows may be better understood and in order that thecontributions they represent to the art may be appreciated. There are,of course, additional features of the disclosure that will be describedhereinafter and which will form the subject of the claims appendedhereto.

BRIEF DESCRIPTION OF THE DRAWINGS

For a detailed understanding of the present disclosure, reference shouldbe made to the following detailed description of the embodiments, takenin conjunction with the accompanying drawings, in which like elementshave been given like numerals, wherein:

FIG. 1 shows an optical displacement device deployed along a wirelineaccording to one embodiment of the present disclosure;

FIG. 2 shows a graph of reflected and transmitted light in a Fabry-Perotinterferometer according to the present disclosure;

FIG. 3 shows a graph of optical responses to a range of incident anglesof light in the Fabry-Perot interferometer according to the presentdisclosure;

FIG. 4 shows a schematic of an optical displacement apparatus accordingto one embodiment of the present disclosure;

FIG. 5 shows a schematic of the divergent light beams produced inanother optical displacement apparatus according to one embodiment ofthe present disclosure;

FIG. 6 shows a schematic of the interferometer and detection array of anoptical displacement apparatus according to one embodiment of thepresent disclosure;

FIGS. 7A and 7B show another example optical displacement sensor with alocked wavelength light source;

FIG. 8 shows another example optical displacement sensor with a lockedwavelength light source;

FIGS. 9A and 9B show an illustrative gravimeter having an opticaldisplacement device with two Fabry-Perot cavities in accordance withembodiments of the present disclosure;

FIG. 10 shows a flow chart of a method for estimating a parameter ofinterest in an earth formation intersected by a borehole according toone embodiment of the present disclosure.

DETAILED DESCRIPTION OF THE DISCLOSURE

The present disclosure relates to devices and methods for estimating aparameter of interest with an optical displacement device usingtechniques for preventing changes in the information generated in theoptical displacement device resulting from temperature or pressurechanges at the optical displacement device. These techniques may includecompensating for the changes at the optical displacement device.

Aspects of the present disclosure include a dual-cavity opticaldisplacement device. In traditional gravimeters, the cavity length mayfluctuate with temperature, producing an offset term which is accountedfor in measurement calculations. By using a common member to support areflective surface for each respective cavity on opposing sides,measurements from the two devices may be related to one another, suchthat effects of temperature changes on one cavity cancel the effects ofthe temperature changes on the other cavity. The cavities of thedual-cavity gravimeter may be designed to be of equal length when theforce on the proof-mass is zero. Thus, the difference in cavity lengthsmay be used as the relevant offset term, which may be configured to besubstantially zero. The member may be used as a proof mass. Someembodiments use reflection only for both cavities. In this case, themirrors attached to the proof-mass spring assembly can be totallyreflective, allowing increased mass for the proof mass. This may improvethe sensor's sensitivity to proof-mass displacement by increasing thefinesse of both cavities.

Aspects of the present disclosure include maintaining a light sourcecontrol element (e.g., a control etalon) in the same environment as anoptical displacement device used to sense information related to theparameter of interest (e.g., a sensor etalon). Optical displacementsensors use a light source that is separated from the sensor by asufficient distance that it is subject to different pressure andtemperature fluctuations. As a result, fluctuations in its wavelengthcaused by environmental factors are uncorrelated with sensor parametersand appear as a noise source. With a laser light source locked to thephysical dimensions of an etalon, the fluctuations are primarily causedby changes in the index-of-refraction of the material between theetalon's mirrors and by changes in length of the cavity. These changescan also occur in the displacement sensor's cavity. By housing criticalcomponents of the laser system with components of the displacementsensor, the laser's locking etalon and the sensor's cavity experiencethe same environment so that changes in the index of refraction anddimensions of the laser's locking etalon are correlated with the samechanges in the sensor's cavity. In this configuration, the thermalcoefficients of the laser's etalon may act to oppose the thermalcoefficients of the sensor; compensating for the sensor's thermalsensitivity.

The present disclosure is susceptible to embodiments of different forms.There are shown in the drawings, and herein will be described in detail,specific embodiments of the present disclosure with the understandingthat the present disclosure is to be considered an exemplification ofthe principles of the present disclosure and is not intended to limitthe present disclosure to that illustrated and described herein.

FIG. 1 shows one embodiment according to the present disclosure whereina cross-section of a subterranean formation 10 in which is drilled aborehole 12 is schematically represented. Suspended within the borehole12 at the bottom end of a carrier such as a wireline 14 is a device ortool 100. The tool 100 may include processor. The wireline 14 may becarried over a pulley 18 supported by a derrick 20. Wireline deploymentand retrieval is performed by a powered winch carried by a service truck22, for example. A control panel 24 interconnected to the tool 100through the wireline 14 by conventional means controls transmission ofelectrical power, data/command signals, and also provides control overoperation of the components in the device 100. Use of a non-rigidcarrier to convey tool 100 is exemplary only. Tool 100 may also beconveyed by a rigid carrier. In some embodiments, the borehole 12 may beutilized to recover hydrocarbons. In other embodiments, the borehole 12may be used for geothermal applications, water production, mining,tunnel construction, or other uses.

In embodiments, the device 100 may be configured to collect informationregarding force or acceleration. The device may also be configured to:(i) actively or passively collect information about the variouscharacteristics of the formation, (ii) provide information about toolorientation and direction of movement, (iii) provide information aboutthe characteristics of the reservoir fluid and/or (iv) evaluatereservoir conditions (e.g., formation pressure, wellbore pressure,temperature, etc.). Exemplary devices may include resistivity sensors(for determining the formation resistivity, dielectric constant and thepresence or absence of hydrocarbons), acoustic sensors (for determiningthe acoustic porosity of the formation and the bed boundary in theformation), nuclear sensors (for determining the formation density,nuclear porosity and certain rock characteristics), nuclear magneticresonance sensors (for determining the porosity and other petrophysicalcharacteristics of the formation), and gravimeters/gravity gradiometers(for estimating formation density). Other exemplary devices may includegyroscopes, magnetometers, accelerometers, and sensors that collectformation fluid samples and determine the properties of the formationfluid, which include physical properties and chemical properties.

Device 100 may be conveyed to a position in operable communication orproximity with a parameter of interest. In some embodiments, device 100maybe conveyed into a borehole 12. The parameter of interest mayinclude, but is not limited to, one of: (i) pressure, (ii) force, and(iii) acceleration. Depending on the operating principle of the device100, the device 100 may be configured to operate under surface andborehole conditions. The ambient temperature in the borehole may exceed120 degrees Celsius (248 degrees Fahrenheit). In other embodiments, adevice 100 may be used at the surface 160.

Device 100 includes an optical displacement sensor in accordance withembodiments of the present disclosure. Optical-displacement sensorsoperate by detecting light reflected by an optical element that changesits reflectivity as a result of displacement (e.g., change in positionover time) of a member in response to an environmental stimulus, such aspressure differential, sound, vibration, etc. The detected light may beconverted into an electrical signal. This signal may be a function ofthe reflectivity of the optical element, and, therefore, a function ofthe stimulus as well. Optical displacement sensor may include an opticalinterferometer.

Optical interferometers have been proven to have outstanding resolutionwhen used as displacement detectors in physical sensing components, suchas microphones, pressure sensors, and accelerometers. One exemplaryoptical interferometer is the Fabry-Perot interferometer, which is alsoknown as an etalon. An etalon may include an optically resonant cavitythat distributes optical energy of an input light signal into areflected signal and a transmitted signal. The ratio of optical energyin the reflected and transmitted signals may depend on the cavity lengthof the optically resonant cavity, which is the spacing between its two,substantially-parallel, partially reflective surfaces and its operatingwavelength, x, (i.e., the wavelength, x, of the light on which theinterferometer operates).

In an etalon, light can be strongly modulated by very small changes inthe cavity length, and these changes can be detected using standardoptical detection approaches that result in a wide dynamic range. Theuse of light beam for the readout is very different from the morestandard approaches that use charged particles, including electrostatic,capacitive, piezoelectric, or piezoresistive techniques. All of thesetechnologies require a close coupling between the transducer design andthe sensor mechanical design. This results in constraints on the sensorthat affect the performance adversely, especially as the size isreduced. An etalon-based displacement sensor having high dynamic rangeand high sensitivity may have many advantages in the field of physicalsensing including reduction in size of the optical interferometertransducer and not adversely interacting with the optical interferencetransducer. This independence between the etalon and the opticalinterference transducer may result in the benefit of a decoupling of thetransducer and the sensor design.

An etalon may be configured to be sensitive to a force or accelerationstimulus by having one surface of the etalon operably connected to asurface of, or disposed on, a movable element. When the element moves inresponse to the stimulus, the cavity length changes and, therefore, sodoes the ratio of optical energy in the reflected and transmittedsignals. As a result, an electrical output signal based on one of thereflected and transmitted signals may be a function of the stimulusincident on the etalon.

The basic operating principal involves the creation of an optical pathwhose length is varied when subjected to an external stimulus, such aspressure or acceleration. An etalon may be used for consideration of thedesign issues in these types of transducers in general. This type ofinterferometer may have two parallel dielectric mirrors that bound anempty cavity. Light that is incident upon the cavity will be partiallytransmitted according to the formula:

$\begin{matrix}{{T = \frac{1}{\left( {1 + {F\;\sin^{2}{\varphi/2}}} \right)^{2}}},} & (1)\end{matrix}$wherein F (finesse) is determined by the reflectance of the two mirrors,

$\begin{matrix}{{F = \frac{4\; R_{0}}{\left( {1 - R_{0}} \right)^{2}}},} & (2)\end{matrix}$wherein R₀ is the reflectance of the mirrors. Herein, it is assumed thatthe reflectance values of the two mirrors are equal. φ=4πnd cos θ/λ isthe phase that is picked up in a wave with wavelength λ as it makes aroundtrip within the cavity of length d and index of refraction n withangle of incidence θ.

A typical response is shown in FIG. 2, where a transmitted wave 210 isreflected as reflected wave 220. The rate of change of the opticalsignal may be on the order of several percent per nanometer of motion.The optical signal can be resolved at a level below 100 parts perbillion (ppb), which translates to the ability of the transducer todetect femtometer-scale changes in the displacement of one mirrorrelative to the other mirror. However, the sensitivity of the transducermay be very low throughout most of the typical operating points as canbe seen in curves 210 and 220. Sensitivity of the transducer may beproportional to the absolute value of the slope of curves 210 and 220.So it may be observed that the slope, and sensitivity, may be very lowwhen the gap d is between about 60.2 micrometers and 60.4 micrometers,and the slope may observed to be higher between about 60.4 micrometersand 60.6 micrometers.

A high resolution sensor may require that the optical cavity length beallowed to vary over many wavelengths. One way to maintain sensitivityover a wide range of cavity lengths is to use multiple beams of lightwhere each beam has a different response to changes in cavity length.The multiple beams of light may exhibit different responses by havingdifferent wavelengths, different angles of incidence, or a combinationthereof.

FIG. 3 shows a chart of curves representing multiple beams on lightusing an exemplary set of different angles of incidence. The angles ofincidence in this example range from 1.00 degrees in curve 310 to 7.00degrees in curve 320. This range of angles is exemplary and illustrativeonly, and other ranges of angles may be used as would be understood byone of skill in the art with the benefit of the present disclosure. Itmay be seen from these curves that more than one beam may be sensitiveto a particular cavity length. These beams may be produced usingtechniques known to those of skill in the art, including, but notlimited to, one or more of: (i) directing multiple light beams toward asingle lens configured to operate in a pupil division mode and (ii)directing multiple light beams toward an individual lensletscorresponding to each of the multiple light beams.

FIG. 4 shows a schematic of one embodiment according to the presentdisclosure. The coherent light beam 410, which may be collimated and/orpolarized, may be split by a beam splitter 415 to generate a referencesignal 420. A cylindrical condenser lens 430 spreads the beam 410 intodivergent beams 480 in the far field. The input tilt mirror 440 sets themean angle of incidence through the etalon 445. In some embodiments, themean angle may range from about 0 degrees to about θ_(max) degrees,where:

$\begin{matrix}{{\theta_{m\;{ax}} < \frac{w}{2d_{m\;{ax}}k_{0}}};} & (3)\end{matrix}$

-   -   w=width of beam;        k ₀=100R(2+R)(1−R ²)⁻¹; and    -   R=Reflectivity of etalon surfaces

It may be understood by one of skill in the art with the benefit of thepresent disclosure that the mean angle may range from about 0 degrees toan angle of such magnitude as causes the light beam to degrade such thatthe light beam may no longer interfere with itself. The light beams 480,each with its own angle of incidence, may be partially transmittedthrough the etalon 445 and may be collimated before reaching a detectorarray 460 by a collimator lens 450. The number of angles of the lightbeams may vary with the finesse of the etalon 445. Finesse is aparameter characterizing an optical cavity and may be a function of thereflectivity of the mirrored surfaces of the etalon 445, herein definedby eqn. (2). Generally, a higher reflectivity may result in a higherdegree of finesse and the higher the number of angles of the light beamsthat may be required in the etalon 445. The detector array 460 mayinclude two or more light sensitive detectors, such as photodiodes. Inthis exemplary embodiment, the detector array 460 includes nine (9)detectors with a spot size of each detector at about 10 micrometers andan array spacing of about 575 micrometers. Detector array 460 maydistribute the detectors linearly. Detector array 460 may include atleast one detector for each of the light beams 480. The detector array460 may be configured to generate electrical signals in response toreceiving energy from the light beams 480. The collimator lens 450 mayinclude a lens array with a plurality of lenses. Condenser lens 470 mayfocus light beam 410 on a reference detector 490. The light beam 410 maybe of any wavelength (infra-red, visible, ultraviolet, etc.) as long asthe corresponding lenses, detectors, and etalon are configured and/orselected to properly respond to the wavelength used. In someembodiments, a second light beam (not shown) at a different wavelengthfrom light beam 410 may be used along with light beam 410.

FIGS. 5 & 6 show a schematic of another embodiment according to thepresent disclosure. Here, coherent light beam 410 passes though acollimator lens 505 and polarizer 508 to a lens 530 that may change theincoming beam 410 into beams 580. Lens 530 may be configured to causeincoming beam 410 to diverge or converge. The beams 580 enter an etalon445, where part of each of the beams 547 is transmitted to a lens array450. Lens 530 is shown as a pair of concave cylindrical lenses, however,this is exemplary and illustrative only, as lens 530 may also be asingle concave lens, a single convex lens, or a pair of convex lenses.Beams 580 may be divergent or convergent. The lens array 450 focuses thebeams on a detector array 460. In some embodiments, a single lens (notshown) may be used in addition to or instead of the lens array 450. Insome embodiments, an optical grating (not shown) may be used in additionto or instead of lens 530. The detector array 460 may be configured togenerate electrical signals in response to receiving energy from thelight beams 580. While there is no reference beam shown, a referencebeam may be added to increase sensitivity for some applications. Thereference beam may be used to cancel noise caused by intensityvariations of the incoming light beam 410. The use of an optionalreference beam may be determined by expertise of those of skill in theart.

In other embodiments, a coherent light beam from a fiber optic sourcemay naturally spread to form divergent beams that may pass through anetalon to a custom lens, including for example, off-axis Fresnel zonesthat may transfer the incident energy of the light beam from eachring-shaped zone and direct it to an individual detector in a detectorarray. The detector array may include at least two detectors (which mayform a linear array) responsive to electromagnetic energy and generateelectrical signals in response to receiving energy from the light beams.In some examples, the array may be a two-dimensional array, which mayinclude a charge coupled device (CCD) such as the type used in digitalcameras.

FIGS. 7A and 7B show another example optical displacement sensor with alocked wavelength light source. The optical displacement device 700(FIG. 7A) includes a light source 702 having its wavelength controlledusing an optical control component. For example, the device 700 mayemploy the closed loop light source control to lock the light source tothe length of a high finesse temperature controlled etalon. FIG. 7Ashows a light source module. FIG. 7B shows a sensor module. The opticaldisplacement sensor includes the light source and sensor as separatemodules, coupled by a polarization maintaining optical fiber to transferlight to the sensor module. Each module contains an interferometer, suchas an etalon. The etalon in the light source module (FIG. 7A) is used tolock the wavelength of the laser diode to its length and supply thesensor (FIG. 7B) with light at a constant wavelength. Its etalonresponds to outside forces by displacing one or both of its mirrors.

Referring to FIG. 7A, the light source module 700 contains a number offree space optical components including a laser diode 702, an isolator,and a closed loop light source control. The closed loop light sourcecontrol includes an optical control component (locking etalon 720), oneor more detectors (photodiode 710′) optically coupled to the opticalcontrol component, and control electronics (laser control module 712)operatively coupled to the at least one detector and laser diode 702.Light source module 700 also includes a thermoelectric cooler (‘TEC’)716 to regulate the temperature of the laser diode 702.

The light source module 700 is configured so that the outputelectromagnetic beam (‘light’) is coupled into a fiber, which in turn iscoupled to a fiber splitter. The fiber splitter is configured to send aportion (e.g., 90 percent) of the light out to the sensor module 701 andanother portion (10 percent) to the locking etalon 720. Inside thermallycontrolled environment 708, the light propagates in free spaceto/from/through the locking etalon 720. Locking etalon 720 ishermetically sealed and slightly canted to minimize issues arising fromstray and unwanted reflectance.

Those skilled in the art, will recognize that the locking etalon couldbe replaced by a gas cell containing a gas with an optical adsorptionline at the wavelength required of the sensor. In this case, the laserwould be locked to the wavelength of the adsorption line. A gas that istypically used for wavelengths near 1550 nm is methane. In this case,the laser is locked to one of the rotational adsorption lines of themethane molecule. Those skilled in the art could select different gasesfor different wavelengths. However, using a gas cell will not compensatefor thermal and index of refraction affects inside the sensor etalon.

FIG. 7B includes an optical displacement sensor 701, which may beimplemented according to various embodiments as described herein oremploy various combinations of elements and components thereof. Thelight from the laser enters through a fiber and then propagates in freespace inside the sensor's temperature controlled environment.

As illustrated in FIG. 7B, the optical displacement sensor 701 includessensor optics 703, detectors 707, and an optical displacement device(sensor etalon 705). Sensor etalon (or measurement etalon) 705 isconfigured to estimate the parameter of interest. Sensor optics 703divide the beam into a reference beam and a beam introduced to thesensor cavity of etalon 705. The optics are configured to send thereference beam to a photodiode to serve as an intensity reference.

In the figure, the beam is shown as expanding through the cavity with afocal point distant from the cavity. It could equally well be focusedinto the cavity with a focal point at the cavity's center. Once the beamhas exited the cavity, eight fractions of it are collected and focusedon an array of sensors, e.g., photodiodes 707. The response of thephotodiodes 707 is received by sensor electronics 709, which may includeone or more processors, along with the reference signal. From there itis processed to yield a value for the optical phase of each beam, asdescribed above.

The etalon in the laser and the etalon in the sensor cavity may be inseparate hermetically sealed housings and regulated to separatetemperatures. The optical fiber between the two modules may cross asignificant distance separating the modules.

Returning to FIG. 7A, in operation, laser control module 712 locks thewavelength of the light source 702 to the length of a high finessetemperature controlled etalon. The index of refraction will changebecause of atmospheric changes as well as with temperature andcomposition of the gas. The temperature is controlled to limit thermalchanges in dimension.

Responsive to light received, photodiodes 710, 710′ send signalsrepresenting information indicative of the respective light received tolaser control module 712. Thus, photodiodes 710, 710′ in cooperationwith laser control module 712 measure the amount of reflected (R) andtransmitted (T) light. The difference (R−T) is used to lock thewavelength of the laser diode to the side of an etalon fringe. Forexample, the wavelength may be about 1550 nm.

The locking occurs by detecting the light reflecting from and passingthrough the etalon. The transmitted light is given by

$\begin{matrix}{{T = \frac{I_{0}}{1 + {F\;\sin^{2}{\delta/2}}}};} & (4)\end{matrix}$I₀ is the intensity of the transmitted light; F is the finesse factor ofthe cavity; and δ is the optical phase of the light in the cavity. Theoptical phase is

$\begin{matrix}{{{\delta = \frac{2\;\pi\; v}{v_{{FSR},L}}};}{{v_{{FSR},L} = \frac{c}{2\; d_{L}n_{L}\cos\;\theta_{L}}};}} & (5)\end{matrix}$Γ_(FSR,L) is the free spectral range of the cavity; c is the speed oflight d_(L) is the length of the cavity; n_(L) is theindex-of-refraction for the media in the cavity; and θ_(L) is the lightbeam's angle of incidence to the cavity. Assuming an ideal cavity,reflected light is given by

$\begin{matrix}\begin{matrix}{R = {I_{0} - T}} \\{= {I_{0}\left( {1 - \frac{1}{1 + {F\;\sin^{2}{\delta/2}}}} \right)}} \\{= {{I_{0}\left( \frac{F\;\sin^{2}{\delta/2}}{1 + {F\;\sin^{2}{\delta/2}}} \right)}.}}\end{matrix} & (6)\end{matrix}$To lock the wavelength, a small offset current applied that isproportional to R−T. This is equivalent to locking at T=R=½. In thatcase, the optical phase must satisfyF sin²δ/2=1.  (7)The stable light source wavelength and frequency must satisfy

$\begin{matrix}{v = {{v_{{FSR},L}\left( {m \pm {\frac{1}{\pi}{\sin^{- 1}\left( \frac{1}{\sqrt{F}} \right)}}} \right)} \equiv {\alpha\; v_{{FSR},L}}}} & (8)\end{matrix}$

Whether the ± sign is applicable depends on which side of the fringe thelight source is locked to. Here m is an integer and is about 7740 for a6 mm cavity and a nominal wavelength of 1550 nm. For a high finessefactor of 100, then inverse sine is about 0.1. The important point isthat the frequency of the locked laser is proportional to the freespectral range of the cavity. It means that it is also proportional tothe index-of-refraction of the media in the cavity and its length.

The sensor receives the input light from the light source. The inputlight passes through an etalon where one of the mirrors is attached to aproof mass and spring and we measure the transmitted light. Assumingperfect light measurement (ignoring for the moment the use of thereference beam to remove relative intensity noise from the measurement),the signal obtained is given by eqn. (4), but the free spectral rangedepends on the sensor cavity.

The transmitted signal from any given channel of the sensor is thengiven by

$\begin{matrix}{{S = \frac{I_{C\; 0}}{1 + {F_{C}\sin^{2}{\delta_{C}/2}}}};} & (9)\end{matrix}$where S is the signal; I_(C0) is the incident intensity; F_(C) is thefinesse of the cavity and δC is the optical phase of the light throughthe cavity. Referring to the optical phase,

$\begin{matrix}{{{\delta_{C} = \frac{2\;\pi\; v}{v_{{FSR},C}}};}{v_{{FSR},C} = {\frac{c}{2\left( {x_{C} - x} \right)n_{C}\cos\;\theta_{C}}.}}} & (10)\end{matrix}$

Here the subscript C indicates the variables of associated with thesensor cavity and x_(C) is the length of the cavity under zero force onthe proof mass. The proof-mass moves a distance x, under the force ofgravity. As is readily apparent,

$\begin{matrix}{{x = \frac{m\; g}{k}},} & (11)\end{matrix}$where m is the mass of the proof-mass and spring assembly; g is thegravitational acceleration; and k is the spring constant. Substitutingthe light frequency from a locked lightsource yields

$\begin{matrix}{\delta_{C} = {\frac{2\;{\pi\alpha}\; v_{{FSR},L}}{v_{{FSR},C}}.}} & (12)\end{matrix}$Applying the expressions for the laser's free spectral range and thecavity's free spectral range, we find

$\begin{matrix}{\delta_{C} = {2\;\pi\;\alpha{\frac{\left( {x_{C} \pm x} \right)n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}.}}} & (13)\end{matrix}$The ± sign depends on the orientation of the cavity with respect togravity. All of the parameters of this equation (with the exception ofα) can be temperature dependent.

The displacement of the proof-mass is dependent upon the gravitationalforce and the spring constant of the spring. The spring constant dependson the mechanical dimensions of the spring and its elastic constants.The thermal constants of dimensions and elastic constants act in a waysuch that the effects counteract. If the temperature fluctuations aresufficiently small and the spring material is chosen properly, then thedisplacement can have a temperature coefficient that is close to zero.However, generally the thermal coefficient will be dominated by theelastic constant's thermal coefficient which is on the order of hundredsof parts per million. It is assumed that the coefficient is a few partsper million.

An approximate expression for the index of refraction of air is

$\begin{matrix}{{{{{n = {1 + {a\frac{P}{\left( {T^{\prime} + 273} \right)}} + {{bH}_{R}\left( {{T^{\prime}}^{2} + 160} \right)}}};}a = {7.86 \times 10^{- 4}{K\left( {k\;{Pa}} \right)}^{- 1}}};}{{b = {1.5 \times 10^{- 11}{K(\%)}^{- 1}}};}} & (14)\end{matrix}$H_(R) is the relative humidity in percent and T′ is the temperature incentigrade. The first term is the ideal gas law inserted into theexpression for the index of refraction as a function of density. Attypical ambient conditions, the first term is approximately 2.7×10⁻⁴. Ormore simply,n=1+ξN;ξ=1.092×10⁻²⁹ m³;N=Number Density(1/m³).  (15)

Because etalons for the optical control component of the locked lightsource and for the optical displacement sensor (e.g., a laser etalon)are separately hermetically sealed at their separate locations, theetalons for the light source and the sensor cavity reside in differenthermetically sealed environments.

The density of a gas in a volume is given by the total number ofmolecules, N_(#) divided by the volume, V. Thus we have

$\begin{matrix}{{\mathbb{d}N} = {{\frac{\partial N}{\partial N_{\#}}{\mathbb{d}N_{\#}}} + {\frac{\partial N}{\partial V}{{\mathbb{d}V}.}}}} & (16)\end{matrix}$

This is equal to zero, since N_(#) is fixed. Thus,

$\begin{matrix}\begin{matrix}{\frac{\mathbb{d}N}{\mathbb{d}T} = {\frac{\partial}{\partial V}\left( \frac{N_{\#}}{V_{0}} \right)\frac{\mathbb{d}V}{\mathbb{d}T}}} \\{= {{- \frac{N_{\#}}{V_{0}^{2}}}\frac{\mathbb{d}}{\mathbb{d}T}{V_{0}\left( {1 + {3{C_{TE}\left( {T - T_{0}} \right)}}} \right)}}} \\{= {{- \frac{N_{\#}}{V_{0}}}3C_{TE}}} \\{= {{- 3}C_{TE}N_{0}}}\end{matrix} & (17)\end{matrix}$where the subscript 0 refers to the volume at T₀, and C_(TE) is thelinear coefficient of thermal expansion. Thus the density is given byN=N ₀(1+3C _(TE)(T−T ₀)),  (18)and the index of refraction is given by

$\begin{matrix}\begin{matrix}{{n = {1 + {\xi\;{N_{0}\left( {1 + {3{C_{TE}\left( {T - T_{0}} \right)}}} \right)}}}};} \\{= {n_{0} + {3\xi\; C_{TE}{{N_{0}\left( {T - T_{0}} \right)}.}}}}\end{matrix} & (19)\end{matrix}$Thus, evaluating this expression for a cavity with a room temperature,room pressure gas, and aluminum container,

$\begin{matrix}{\frac{\delta\; n_{0}}{n_{0}} \approx {3\xi\; C_{TE}N_{0}\delta\; T} \equiv {C_{T}\delta\; T} \approx {3 \times \left( {1.1 \times {10^{- 29}\left\lbrack m^{+ 3} \right\rbrack}} \right)\left( {23 \times {10^{- 6}\left\lbrack K^{- 1} \right\rbrack}} \right)\left( {2.4 \times {10^{25}\left\lbrack m^{- 3} \right\rbrack}} \right)\delta\; T} \approx {\left( {1.85 \times {10^{- 8}\left\lbrack K^{- 1} \right\rbrack}} \right)\delta\; T}} & (20)\end{matrix}$Thus, regarding index of refraction, by regulating the temperature tochanges of fewer than 0.5 degrees (and the housing has a C_(TE) of lessthan ten ppm), a separate hermetically sealed enclosure for laser andsensor may used with acceptable results. However, with regards toangular stability, the angle of the light beams entering the cavity issmall, less than 10-15 degrees. Using a Taylor expansion in temperature,we havecos θ≈cos θ₀(1−C _(T)θ₀ ² δT).  (21)In this case, C_(T) may be 5-15 ppm, and the angle less than 0.2radians. Thus, acceptable results may be achieved if the temperature ismaintained within 1 millikelvin.

However, in light of the difference in environment between the sensormodule and the light source module, achieving the desired degree ofprecision becomes problematic. Repeating equation (10),

$\begin{matrix}{\delta_{C} = {2\pi\;\alpha{\frac{\left( {x_{C} \pm x} \right)n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}.}}} & (22)\end{matrix}$

Substituting in all the thermal coefficients and keeping only the linearterms,

$\begin{matrix}{{\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \approx {\left( {\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}\left\lbrack {1 + {\left( {C_{T,n_{c}} + C_{T,\theta_{C}} + {{\frac{x_{C}}{\left( {x_{C} \pm x} \right)}C_{T,x_{C}}} \pm {\frac{x}{\left( {x_{C} \pm x} \right)}C_{T,x}}}} \right)\delta\; T_{C}} - {\left( {C_{T,n_{L}} + C_{T,\theta_{L}} + C_{T,d_{L}}} \right)\delta\; T_{L}}} \right\rbrack}} & (23)\end{matrix}$where the zero subscript refers the value at the mean temperature ofeach etalon. Given the conclusion of Index of Refraction section, thetemperature coefficients of the index of refraction may be ignored.

$\begin{matrix}{{\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \approx {\left( {\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}\left\lbrack {1 + {\left( {C_{T,\theta_{C}} + {{\frac{x_{C}}{\left( {x_{C} \pm x} \right)}C_{T,x_{C}}} \pm {\frac{x}{\left( {x_{C} \pm x} \right)}C_{T,x}}}} \right)\delta\; T_{C}} - {\left( {C_{T,\theta_{L}} + C_{T,d_{L}}} \right)\delta\; T_{L}}} \right\rbrack}} & (24)\end{matrix}$

By inspection we can write down the error associated with thefluctuation in temperature.

$\begin{matrix}{\frac{\sigma_{\delta_{C}}^{2}}{\left( {2\pi\;\alpha} \right)^{2}} = {\left( {\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}^{2}\left\lbrack {{\left( {C_{T,\theta_{C}} + {{\frac{x_{C}}{\left( {x_{C} \pm x} \right)}C_{T,x_{C}}} \pm {\frac{x}{\left( {x_{C} \pm x} \right)}C_{T,x}}}} \right)\delta\; T_{C}^{2}} + {\left( {C_{T,\theta_{L}} + C_{T,d_{L}}} \right)^{2}\delta\; T_{L}^{2}}} \right\rbrack}} & (25)\end{matrix}$Thus, the errors in the laser add to the errors in the sensor becausefluctuations in the temperature at each module are uncorrelated.

FIG. 8 shows another example optical displacement sensor with a lockedwavelength light source. The optical displacement sensor includes anoptical displacement device for measurement of the parameter ofinterest, and a closed loop light source control including an opticalcomponent at the optical displacement device. The closed loop lightsource control is configured to compensate for changes in temperature atthe optical displacement device by adjusting the at least one lightsource.

In the apparatus of FIG. 8, the temperature and pressure sensitivedevices (e.g., the locking etalon and sensor etalon) of the laser systemand the sensor are combined in a single housing 830. This means that thetemperature of the laser etalon and sensor etalon are equal; their gaspressures are equal; and their gas compositions are equal. Furthermore,any fluctuations in the properties of the cavities are correlated. Thehousing 830 may also be hermetically sealed. In some embodiments, thecomponents are configured for free space transmission of the beam. Thus,polarization maintaining fiber may be altogether eliminated, therebyavoiding fluctuations in laser light polarization inherent in fiber.

The optical displacement device 800 includes a number of free spaceoptical components including a laser diode 802 and a closed loop lightsource control. The closed loop light source control includes an opticalcontrol component (locking etalon 820), one or more detectors(photodiode 810′) optically coupled to the optical control component,and control electronics (laser control module 812) operatively coupledto the at least one detector and laser diode 802. The opticaldisplacement device 800 also includes a thermoelectric cooler (‘TEC’)816 to regulate the temperature of the laser diode 802. The opticaldisplacement device 800 further includes sensor optics 803, detectors807, and an optical displacement device (sensor etalon 805).

For optical displacement device 800, the sensor cavity temperature andthe laser locking etalon temperatures may be equated.

$\begin{matrix}{{\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \approx {\left( {\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}\left( {1 + {\left( {C_{T,\theta_{C}} + {{\frac{x_{C}}{\left( {x_{C} \pm x} \right)}C_{T,x_{C}}} \pm {\frac{x}{\left( {x_{C} \pm x} \right)}C_{T,x}}} - C_{T,\theta_{L}} - C_{T,d_{L}}} \right)\delta\; T_{C}}} \right)}} & (26)\end{matrix}$

The sensor cavity and the laser etalon may be manufactured from matchingmaterials, resulting in their coefficients of thermal expansion beingequal, other than differences caused by the pieces being exposed toslightly manufacturing processes (different batches). That is, theoptical phase of the sensor signals are independent of the thermalexpansion of the both the laser etalon and sensor cavity provided thecoefficients of thermal expansion are matched. Thus,

$\begin{matrix}{{\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \approx {\left( {\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}\left( {1 + {\left( {C_{T,\theta_{C}} - {C_{T,\theta_{L}} \pm {\frac{x_{C}}{\left( {x_{C} \pm x} \right)}\left( {C_{T,x} - C_{T,x_{C}}} \right)}}} \right)\delta\; T_{C}}} \right)}} & (27)\end{matrix}$This eliminates the error caused by thermal fluctuations in the lengthof the laser etalon and sensor cavity lengths. Furthermore, if thethermal coefficient of the etalon is also matched to that of the spring,there is a further reduction in the temperature sensitivity of thedevice. The optical phases of the sensor signals are independent of thethermal coefficient of the spring constant of the proof-mass springassembly. In any case, holding the laser wavelength over the long termis no longer necessary. In the ideal case, the above equation reduces to

$\begin{matrix}{{{\frac{n_{C}\cos\;\theta_{C}}{d_{L}n_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \approx {\left( {\frac{\cos\;\theta_{C}}{d_{L}\cos\;\theta_{L}}\left( {x_{C} \pm x} \right)} \right)_{0}\left( {1 + {\left( {C_{T,\theta_{C}} - C_{T,\theta_{L}}} \right)\delta\; T_{C}}} \right)}},} & (28)\end{matrix}$and the error depends only on the fluctuations in a single temperature(e.g., the temperature at the sensor etalon).

FIGS. 9A and 9B show an illustrative gravimeter having an opticaldisplacement device with two Fabry-Perot cavities in accordance withembodiments of the present disclosure. Referring to FIG. 9A, the opticaldisplacement device 900 is configured to receive at least oneelectromagnetic beam 902 and includes a first reflective surface 904; asecond reflective surface 906; a third reflective surface 908; and afourth reflective surface 910. The third reflective surface 908 may befixed with respect to the second reflective surface 906 and the fourthreflective surface 910 may be fixed with respect to the first reflectivesurface 904.

A lower optical cavity is formed by third reflective surface 908 andfourth reflective surface 910. The third reflective surface 908 may beattached to the frame 920 via a spring 922 with a spring constant k. Anupper optical cavity is formed by first reflective surface 904 andsecond reflective surface 906. The second reflective surface 906 mayalso be attached to the frame 920 via a spring 922 with a springconstant k. The second and third reflective surfaces may be formed as atwo-sided mirror 907 by applying a coating to opposing sides of a solidmember. The solid member may function as the proof mass. Thisconfiguration results in a first variable gap between the firstreflective surface and the second reflective surface; and a secondvariable gap between the third reflective surface and the fourthreflective surface.

Supposing that the output of identical sensors may be generallyexpressed ass=ag _(z) +b  (29)where s is the output, g_(z) is the component of gravity perpendicularto the sensor's sensitive axis, and a and b are the linear constants ofthe sensor that may be time and temperature dependent. In opticaldisplacement device 900, where the two sensors are anti-parallel, theoutput of the sensors may be expressed respectively ass′=a′g _(z) +b′,ands==−ag _(z) +b  (30)

In the case of identical sensors, the linear constants are identical,ands′−s=2ag _(z),ands′+s=2b.  (31)

Thus the differential signal has a zero offset and the total signal isjust twice the value of each sensor's offset. In the case of the opticalsensor, after sufficient signal processing, the produced signal is thelength of the Fabry-Perot cavity. The two-sided mirror 907 deflects whena force is applied by an amount x given bykx=mg _(z)  (32)where m is the mass of the two-sided mirror/spring system. If the cavitylength is x₀ when zero force is applied, then the signal measured by thesystem iss=x ₀ −x=x ₀−(m/k)g.  (33)

Relating this back to equation (29), we can identifya=−(m/k);b=x ₀  (34)

Light reflected between the first and second reflective surfaces may bedescribed in a related way, such that the second cavity respondsidentically to the first cavity except that displacement of the mirroris in the opposite direction. Thus, we haves′=x′ ₀ +x=x′ ₀+(m/k)g.  (35)

The cavities are of different lengths. However, the constant for thelinear term is identical to the second cavity. Thus the signal sum anddifference may be expressed ass′−s=2(m/k)g _(z)+(x′ ₀ −x ₀),ands′+s=2(x′ ₀ +x ₀).  (36)

Inverting eqn. (36) for the component of gravity, we find the estimatefor the gravity component as

$\begin{matrix}{{{{\hat{g}}_{z} = {{\frac{\omega_{0}^{2}\Delta\; s}{2} - \frac{\omega_{0}^{2}\Delta\; x_{0}}{2}} = {{\alpha_{d}\Delta\; s} - \beta_{d}}}};}{{{\Delta\; s} = {s^{\prime} - s}};{{\Delta\; x_{0}} = {x_{0}^{\prime} - x_{0}}};{\omega_{0} = {\sqrt{k/m}.}}}} & (37)\end{matrix}$

Here, α and β are used for the coefficients of the gravity estimator.The equivalent equations for the lower and upper cavities respectivelyareĝ _(z)=ω₀ ² s+ω ₀ ² x ₀ =αs+β;ĝ _(z)=ω₀ ² s′+ω ₀ ² x ₀ ′=−α′s′+β′.  (38)

When comparing the baseline offset terms (β) in eqns. (37) and (38), itis readily apparent that the offset term for the differentialmeasurement can, in principle, be set to zero by making the initialcavity lengths of the two cavities equal, thereby approximating azero-length spring. In practice, the term may be reduced to thetolerance to which the cavities lengths are manufactured.

Using currently available technology, the apparatus may be configured tomaintain a constant etalon temperature to ±1 mK and the cavities have atolerance of 25 micrometers. However, the cavities in a dual cavitysystem may be designed to be of equal length when the gravitationalforce on the proof-mass is zero, resulting in minimum value of offsetthat is limited only by manufacturing tolerance. For a dual cavitysystem, the difference in cavity lengths (and thus, the relevant offsetterm) could be approximately 10 nanometers or less. So the processreduces the offset term by a factor of 10⁻⁴ over a single cavity sensorwith a 100 micrometer cavity. In addition, if the spacers for eachcavity were manufactured from the same material and lot, the thermalexpansion of each might only differ by 1 percent (δ∈_(x0)≈0.01 ∈_(x0)).

Moreover, the offset term in eqn. (37) is much smaller than those ineqn. (38) because the tolerance is usually much smaller than the actualdimension. If t is the tolerance then the offset term may be reduced byβ_(d) /β=t/x ₀  (39)

The temperature dependence of the linear term depends on the springconstant k and thermal expansion of the cavities. The change in offsetisδβ_(d) /δT=β _(d)(∈_(Y)+∈_(L)+∈_(x0))+(ω₀ ² x ₀′/2)δ∈_(x0),  (40)where ∈ is the temperature coefficient and the subscript indicates thecorresponding property. The subscript Y is for Young's modulus, L is forspring length, and x₀ is for cavity length. The first term is theregular expression for the temperature dependence of the offset term andis the same as it would be for an individual measurement. The secondterm indicates that the coefficients of expansion for the cavity lengthsmay not be identical. When compared to the thermal dependence of asingle cavity device, we find

$\begin{matrix}{{\delta\;{\beta_{d}/\delta}\; T} \approx {{\beta\left\lbrack {{\frac{t}{2x_{0}^{\prime}}\left( {ɛ_{Y} + ɛ_{L} + ɛ_{x\; 0}} \right)} + {\delta\; ɛ_{x\; 0}}} \right\rbrack}{\operatorname{<<}\delta}\;{\beta/\delta}\;{T.}}} & (41)\end{matrix}$

The thermal dependence of the proportionality constant isδα_(d) /δT=α _(d)(∈_(Y)+∈_(L)),  (42)and is identical to a single cavity device. The thermal coefficient forYoung's modulus for many materials is −100 ppm while the coefficient ofthermal expansion is on the order of a few parts per million. Howeverthere are several materials for which eqn. (42) is approximately zero.Two of these have coefficients of thermal expansion of 4.5 to 6.5 ppmand 8.0 ppm from room temperature to 300 C. For these to satisfy eqn.(42), the thermal coefficient for Young's modulus is on the same orderbut negative. When eqn. (42) is zero,δβ_(d) /δT=β _(d)∈_(x0)+(ω₀ ² x ₀′/2)δ∈_(x0).  (43)

For a nominal cavity length of 100 um, the thermal coefficient for theestimate of gravity is

$\begin{matrix}{{\frac{\mathbb{d}}{\mathbb{d}T}{\hat{g}}_{z}} = {\frac{\mathbb{d}}{\mathbb{d}T}\left( {{\alpha_{d}\Delta\; s} + \beta_{d}} \right)}} \\{= \left( {{{ɛ\left( \alpha_{d} \right)}\alpha_{d}\Delta\; s} + {\beta_{d\;}ɛ_{x_{0}}} + {\frac{\omega_{0}^{2}x_{0}^{\prime}}{2}\delta\; ɛ_{x_{0}}}} \right)} \\{\approx {\left( {\beta_{d} + {0.01\;\frac{\omega_{0}^{2}x_{0}^{\prime}}{2}}} \right)ɛ_{x_{0}}}} \\{\approx {\left( {5 \times 10^{- 6}} \right)\frac{\left( {2\pi\; 100} \right)^{2}}{2}\left( {{10 \times 10^{- 9}} + {0.01\left( {100 \times 10^{- 6}} \right)}} \right)}} \\{= {9.8 \times 10^{- 7}\mspace{14mu} m\text{/}s^{2}\text{-}K}}\end{matrix}$Thus we haveσ_(ĝz)=(9.86×10⁻⁷ m/s² −K)σ_(T)≈1×10⁻⁹ m/s²  (44)

For a single-sided sensor with a nominal cavity length of 100 um,multiply by 4 and we have σ_(ĝz)≈0.1 uGal. Further improvements could bemade by using zerodur or other low thermal expansion materials.

Assuming that the sensitive axis of the differential sensor is parallelto gravity (the ‘up’ orientation), and that a rotation of the sensor 180degrees about an axis perpendicular to the sensitive axis is the ‘down’position,Δs ₊=(s′−s)₊=2ω₀ ⁻² g+Δx ₀,andΔs ⁻=(s′−s)⁻=−2ω₀ ⁻² g+Δx ₀,  (45)thenΔs ₊ −Δs ⁻=−4ω₀ ⁻² g;andΔs ₊ +Δs ⁻=2Δx ₀.  (46)Combining this with the sum of the signals, it is possible to solve forthe initial lengths of the two cavities.2

s

_(±) =x′ ₀ +x ₀;2

Δs _(±)

=2(x′ ₀ −x ₀)  (47)Solving for the cavity lengths,

$\begin{matrix}{{{x_{0}^{\prime} = {\left\langle s \right\rangle_{\pm} + {\frac{1}{2}\left\langle {\Delta\; s_{\pm}} \right\rangle}}};}{x_{0} =}{\left\langle s \right\rangle_{\pm} - {\frac{1}{2}{\left\langle {\Delta\; s_{\pm}} \right\rangle.}}}} & (48)\end{matrix}$

In some embodiments, it is further possible to apply a smallacceleration to the frame with a frequency below the resonant frequencyof the proof-mass and spring assembly, for example, by using an actuator(e.g., mechanical, piezoelectric, magnetic, etc.). The acceleration maybe modeled asa(t)=a ₀ cos ωt;ω<<ω ₀  (49)Then in addition the possibly large component of gravity we have a smallamplitude signal. At the minimum and maximum amplitudes of the smallsignal, we haveΔs _(+a)=2ω₀ ⁻²(g _(z) +a ₀)+Δx ₀,andΔs _(−a)=2ω₀ ⁻² g(g _(z) −a ₀)+Δx ₀.  (50)ThenΔs _(+a) −Δs _(−a)=−4ω₀ ⁻² a ₀.  (51)

Thus, in calibrating the dual cavity device, a greatly reduced offsetterm may be achieved in relation to conventional optical displacementdevices.

FIG. 9B shows another gravimeter having an optical displacement devicewith two Fabry-Perot cavities in accordance with embodiments of thepresent disclosure. The optical displacement device 901 is configured toreceive at least one separate corresponding electromagnetic beam 951,951′ for each cavity 990, 992. Each cavity 990, 992 is configured tooutput a separate corresponding electromagnetic beam (953, 953′)eliminating any need for a transparent or translucent mirrorincorporated in the proof mass 994. Thus, the two sided mirror may beopaque. In one example, the central member may be manufactured fromTungsten or a similarly dense material. The mass of the proof mass,therefore, may be significantly greater (keeping the dimensions fixed)than in the embodiment of FIG. 9A, resulting in greater precision ingravitational measurements. However, the use of separate inputs andoutputs for each cavity may increase design complexity and requireadditional space, thus making the choice between embodiments accordingto FIG. 9A and FIG. 9B application specific and dependent upon a numberof design considerations.

Input beams 951, 951′ and output beams 953, 953′ may be configured usinglight input components 950, 956 (which may be light sources,intermediate optics, or optical fiber); and light output detectioncomponents (which may be light sensors, intermediate optics, or opticalfiber); and associated optical components 960-974.

FIG. 10 shows of flow chart of a method 1000 for estimating a parameterof interest in an earth formation intersected by a borehole according toone embodiment of the present disclosure. The method 1000 may includeusing devices 100, 800, 900, 901, and so on. In step 1010, an opticaldisplacement device is conveyed into the borehole 12. For example, theoptical displacement device may be conveyed using conveyance device (orcarrier) 14. The optical displacement device may be incorporated as aninstrument in a downhole logging tool.

In step 1020, the optical displacement device is used to generateinformation from the optical displacement device relating to relativemotion between two or more reflective surfaces of the opticaldisplacement device that is indicative of the parameter of interest. Forexample, a plurality of light beams 880 may be transmitted into anetalon 445 and partially transmitted from the etalon 445 to a detectorarray 860. An external stimulus (such as force or acceleration) causes adisplacement in one of the mirrored surfaces of the etalon 445, whichchanges the cavity length of the etalon 445. Electrical signalsgenerated by the detector array 860 due to the partially transmittedlight beams may be altered as a result of change in cavity length of theetalon 445. The signals may carry (embody) the information. An externalstimulus may be estimated based on the change in the electrical signalsgenerated by the detector array 860. In some embodiments, the externalstimulus estimation may also use a reference signal generated by areference detector. In some embodiments, step 1020 may be performed witha single light beam that is moved through a range of angles of incidenceover a period of time, such that the electrical signals generated may beproduced sequentially.

In step 1030, the method includes preventing changes in the informationresulting from changes at the optical displacement device in at leastone of i) temperature, or ii) pressure, by compensating for the changesat the optical displacement device. Compensating may be carried out byadjusting at least one light source generating an electromagnetic beam(at least partly received by the optical displacement device, theadjusting responsive to information relating to a control opticaldisplacement device (e.g., locking etalon 720) at the opticaldisplacement device. For example, adjusting the at least one lightsource may be carried out using at least one detector optically coupledto the control optical displacement device; and control electronicsoperatively coupled to the at least one detector and the at least onelight source and configured to provide closed loop control of the atleast one light source using information from the at least one detector.Alternatively, compensating may be carried out by using an opticaldisplacement device including a first reflective surface; a secondreflective surface; a third reflective surface fixed with respect to thesecond reflective surface; a fourth reflective surface fixed withrespect to the first reflective surface; a first variable gap betweenthe first reflective surface and the second reflective surface; and asecond variable gap between the third reflective surface and the fourthreflective surface; and by configuring the optical displacement devicesuch that the difference between the first variable gap and the secondvariable gap is substantially zero while the apparatus is subject tonominal conditions.

Displacement as used herein means change in position over time.Displacement may include relative displacement (e.g., displacement of amember relative to a sensor, another member, or the rest of the device)or absolute displacement (e.g., displacement relative to the earth).

The term “carrier” as used herein means any device, device component,combination of devices, media and/or member that may be used to convey,house, support, or otherwise facilitate the use of another device,device component, combination of devices, media and/or member. Exemplarynon-limiting carriers include drill strings of the coiled tube type, ofthe jointed pipe type, and any combination or portion thereof. Exemplarynon-limiting conveyance devices include drillstrings of the coiled tubetype, of the jointed pipe type and any combination or portion thereof.Other conveyance device examples include casing pipes, wirelines, wireline sondes, slickline sondes, drop shots, downhole subs, BHA's,drillstring inserts, modules, internal housings and substrate portionsthereof, and self-propelled tractors.

The term “information” as used above includes any form of information(analog, digital, EM, printed, etc.). Herein, the term “information” mayinclude one or more of: raw data, processed data, and signals. The term“processor” herein includes, but is not limited to, any device thattransmits, receives, manipulates, converts, calculates, modulates,transposes, carries, stores or otherwise utilizes information. Inembodiments, the processor may be configured to include resident memory(and/or peripherals) operatively coupled to the processor, so as to beaccessible to the processor for executing programmed instructions. Inseveral non-limiting aspects of the disclosure, a processor may beembodied as a computer that executes programmed instructions forperforming various methods. The processor may execute instructionsstored in computer memory accessible to the processor, or may employlogic implemented as field-programmable gate arrays (‘FPGAs’),application-specific integrated circuits (‘ASICs’), other combinatorialor sequential logic hardware, and so on.

By an offset value of “substantially zero” for the difference in cavitylength under nominal conditions, it is meant a distance of zero allowingfor manufacturing tolerances currently practical for Fabry-Perot etalonsmanufactured for downhole use, examples of such a distance including,for example, less than 100 nm, less than 50 nm, less than 20 nm, lessthan 10 nm, less than 5 nm, less than 3 nm, and so on, down to andincluding zero distance. Herein, environmental conditions “at” acomponent of a sensor refer to conditions at a location sufficientlyclose to the component such that any differences in conditions areindistinguishable on the use of the sensor in estimating a parameter ofinterest of the borehole, examples of such a distance including, forexample, less than 1 meter, less than 20 centimeters, less than 10centimeters, less than 5 centimeters, less than 3 centimeters, and soon, down to and including zero distance. Temperature at the opticaldisplacement device may also be defined as temperature at a locationsufficiently close to the optical displacement device so as to deviatefrom the environmental temperature of the sensor cavity of the opticaldisplacement device by less than 1 percent, 0.5 percent, 0.2 percent,0.1 percent, 0.01 percent, or less; or a temperature at a locationsufficiently close to the optical displacement device so as to deviatefrom the environmental temperature of the sensor cavity such thatmeasurement error of the sensor due to offset is less than 1, 0.5, 0.2,0.1, or 0.01 percent.

Certain embodiments of the present disclosure may be implemented with ahardware environment that includes a processor, an information storagemedium, an input device, processor memory, and may include peripheralinformation storage medium. The hardware environment may be in the well,at the rig, or at a remote location. Moreover, the several components ofthe hardware environment may be distributed among those locations. Theinput device may be any data reader or user input device, such as datacard reader, keyboard, USB port, etc. The information storage mediumstores information provided by the detectors. Information storage mediummay include any non-transitory computer-readable medium for standardcomputer information storage, such as a USB drive, memory stick, harddisk, removable RAM, EPROMs, EAROMs, flash memories and optical disks orother commonly used memory storage system known to one of ordinary skillin the art including Internet based storage. Information storage mediumstores a program that when executed causes information processor toexecute the disclosed method. Information storage medium may also storethe formation information provided by the user, or the formationinformation may be stored in a peripheral information storage medium,which may be any standard computer information storage device, such as aUSB drive, memory stick, hard disk, removable RAM, or other commonlyused memory storage system known to one of ordinary skill in the artincluding Internet based storage. Information processor may be any formof computer or mathematical processing hardware, including Internetbased hardware. When the program is loaded from information storagemedium into processor memory (e.g. computer RAM), the program, whenexecuted, causes information processor to retrieve detector informationfrom either information storage medium or peripheral information storagemedium and process the information to estimate a parameter of interest.Information processor may be located on the surface or downhole.

While the disclosure has been described with reference to exemplaryembodiments, it will be understood that various changes may be made andequivalents may be substituted for elements thereof without departingfrom the scope of the disclosure. In addition, many modifications willbe appreciated to adapt a particular instrument, situation or materialto the teachings of the disclosure without departing from the essentialscope thereof. Therefore, it is intended that the disclosure not belimited to the particular embodiment disclosed as the best modecontemplated for carrying out this disclosure, but that the disclosurewill include all embodiments falling within the scope of the appendedclaims.

While the foregoing disclosure is directed to the one mode embodimentsof the disclosure, various modifications will be apparent to thoseskilled in the art. It is intended that all variations within the scopeof the appended claims be embraced by the foregoing disclosure.

I claim:
 1. An apparatus for estimating a parameter of interest,comprising: an optical displacement sensor comprising: an opticaldisplacement device configured to receive a plurality of electromagneticbeams from a plurality of light inputs, the optical displacement devicecomprising: a first reflective surface; a middle member comprising: asecond reflective surface; a third reflective surface fixed with respectto the second reflective surface; and an opaque proof mass; a fourthreflective surface fixed with respect to the first reflective surface; afirst variable gap between the first reflective surface and the secondreflective surface receiving a first beam of the plurality of beams froma first light input of the plurality of light inputs; and a secondvariable gap between the third reflective surface and the fourthreflective surface receiving a second beam of the plurality of beamsfrom a second light input of the plurality of light inputs; a firstdetector array configured to receive a part of the first beam; and asecond detector array configured to receive a part of the second beam.2. The apparatus of claim 1, wherein a relative motion between the firstreflective surface and the second reflective surface and a relativemotion between the third reflective surface and the fourth reflectivesurface are each indicative of the parameter of interest.
 3. Theapparatus of claim 1, wherein the optical displacement device comprises:an outer member comprising the first reflective surface and the fourthreflective surface.
 4. The apparatus of claim 3, wherein the outermember comprises a proof mass.
 5. The apparatus of claim 1, wherein theoptical displacement device is configured such that the differencebetween the first variable gap and the second variable gap issubstantially zero while the apparatus is subject to nominal conditions.6. The apparatus of claim 5, wherein nominal conditions comprise atleast one of i) reference gravity; ii) zero force; iii) zeroacceleration; iv) zero pressure.
 7. The apparatus of claim 1, whereinthe first light input is on a first side of the middle member and thesecond light input is on a second side of the middle member opposite thefirst side.
 8. A method for estimating a parameter of interest in anearth formation intersected by a borehole, the method comprising:conveying an optical displacement device in the borehole; generatinginformation from the optical displacement device relating to relativemotion between two or more reflective surfaces of the opticaldisplacement device that is indicative of the parameter of interest; andwherein the optical displacement device comprises: an opticaldisplacement sensor comprising: an optical displacement deviceconfigured to receive a plurality of electromagnetic beams from aplurality of light inputs, the optical displacement device comprising: afirst reflective surface; a middle member comprising: a secondreflective surface; a third reflective surface fixed with respect to thesecond reflective surface; and an opaque proof mass; a fourth reflectivesurface fixed with respect to the first reflective surface; a firstvariable gap between the first reflective surface and the secondreflective surface receiving a first beam of the plurality of beams froma first light input of the plurality of light inputs; and a secondvariable gap between the third reflective surface and the fourthreflective surface receiving a second beam of the plurality of beamsfrom a second light input of the plurality of light inputs; a firstdetector array configured to receive a part of the first beam; and asecond detector array configured to receive a part of the second beam.9. The method of claim 8, comprising preventing changes in theinformation resulting from changes at the optical displacement device inat least one of i) temperature, or ii) pressure, by configuring theoptical displacement device such that the difference between the firstvariable gap and the second variable gap is substantially zero while theoptical displacement device is subject to nominal conditions.